Optimal. Leaf size=55 \[ -\frac {\csc ^4(c+d x)}{4 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^2(c+d x)}{2 a^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 43} \[ -\frac {\csc ^4(c+d x)}{4 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^2(c+d x)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^5}-\frac {2 a}{x^4}+\frac {1}{x^3}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^2(c+d x)}{2 a^2 d}+\frac {2 \csc ^3(c+d x)}{3 a^2 d}-\frac {\csc ^4(c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 38, normalized size = 0.69 \[ \frac {\csc ^4(c+d x) (8 \sin (c+d x)+3 \cos (2 (c+d x))-6)}{12 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 57, normalized size = 1.04 \[ \frac {6 \, \cos \left (d x + c\right )^{2} + 8 \, \sin \left (d x + c\right ) - 9}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 36, normalized size = 0.65 \[ -\frac {6 \, \sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right ) + 3}{12 \, a^{2} d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 39, normalized size = 0.71 \[ \frac {-\frac {1}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}+\frac {2}{3 \sin \left (d x +c \right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 36, normalized size = 0.65 \[ -\frac {6 \, \sin \left (d x + c\right )^{2} - 8 \, \sin \left (d x + c\right ) + 3}{12 \, a^{2} d \sin \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 36, normalized size = 0.65 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^2}{2}-\frac {2\,\sin \left (c+d\,x\right )}{3}+\frac {1}{4}}{a^2\,d\,{\sin \left (c+d\,x\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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